Albert Einstein is perhaps the most famous scientist of this century. One of his most wellknown accomplishments is the formula Despite its familiarity, many people don't really understand what it means. We hope this explanation will help!

One of Einstein's great insights was to realize that matter and energy are really different
forms of the same thing. Matter can be turned into energy, and energy into matter. For example, consider a simple hydrogen
atom, basically composed of a single proton. This subatomic particle has a mass of
0.000 000 000 000 000 000 000 000 001 672 kgThis is a tiny mass indeed. But in everyday quantities of matter
there are a lot of atoms! For instance, in one kilogram of pure water, the mass of hydrogen atoms amounts to just slightly
more than 111 grams, or 0.111 kg. Einstein's formula tells us the amount of energy this mass would be equivalent
to, if it were all suddenly turned into energy. It says that to find the energy, you multiply the mass
by the square of the speed of light, this number being 300,000,000 meters per
second (a very large number):
= 0.111 x 300,000,000 x 300,000,000 =
10,000,000,000,000,000 Joules  This is an incredible amount
of energy! A Joule is not a large unit of energy ... one Joule is about the energy released when you drop a textbook to the
floor. But the amount of energy in 30 grams of hydrogen atoms is equivalent to burning hundreds
of thousands of gallons of gasoline!
If you consider all the energy in the full kilogram of water, which also contains oxygen atoms, the total energy equivalent
is close to 10 million gallons of gasoline! Can all this energy really be released? Has it ever been?
The
only way for ALL this energy to be released is for the kilogram of water to be totally annhilated. This process involves
the complete destruction of matter, and occurs only when that matter meets an equal amount of antimatter ...
a substance composed of mass with a negative charge. Antimatter does exist; it is observable as single subatomic particles
in radioactive decay, and has been created in the laboratory. But it is rather shortlived (!), since it annihilates itself
and an equal quantity of ordinary matter as soon as it encounters anything. For this reason, it has not yet been made in measurable
quantities, so our kilogram of water can't be turned into energy by mixing it with 'antiwater'. At least, not yet.
Another
phenomenon peculiar to small elementary particles like protons is that they combine. A single proton forms the nucleus of
a hydrogen atom. Two protons are found in the nucleus of a helium atom. This is how the elements are formed ... all the way
up to the heaviest naturally occuring substance, uranium, which has 92 protons in its nucleus. It is possible to make two
free protons (Hydrogen nuclei) come together to make the beginnings of a helium nucleus. This requires that the protons be
hurled at each other at a very high speed. This process occurs in the sun, but can also be replicated on earth with lasers, magnets, or in the center of an atomic bomb. The process is called nuclear fusion. What makes it interesting is that when the two protons are forced to combine, they don't need as much of their energy
(or mass). Two protons stuck together have less mass than two single separate protons! When the protons are forced
together, this extra mass is released ... as energy! Typically this amounts to about 0.7% of the total mass, converted to
an amount of energy predictable using the formula .
Elements heavier than iron are unstable. Some of them are very unstable! This means that their nuclei, composed
of many positively charged protons, which want to repel from each other, are liable to fall apart at any moment! We
call atoms like this radioactive. Uranium, for example, is radioactive. Every second, many of the atoms in a chunk
of uranium are falling apart. When this happens, the pieces, which are now new elements (with fewer protons) are LESS massive
in total than the original uranium atoms. The extra mass disappears as energy ... again according to the formula ! This process is called nuclear fission.
Both these nuclear reactions release a small portion of the mass involved
as energy. Large amounts of energy! You are probably more familiar with their uses. Nuclear fusion is what powers a
modern nuclear warhead. Nuclear fission (less powerful) is what happens in an atomic bomb (like the ones used against Japan in WWII), or in a nuclear power plant.
Albert Einstein was able to see where an
understanding of this formula would lead. Although peaceful by nature and politics, he helped write a letter to the President
of the United States, urging him to fund research into the development of an atomic bomb ... before the Nazis or Japan
developed their own first. The result was the Manhatten Project, which did in fact produce the first tangible evidence of
... the atomic bomb! 
Hear Albert Einstein state his famous theorem: download an mp3 (104k, zipped) here.
See also: The Birth of Atoms  Einstein's Theory of Relativity
Physics  Science Page  Worsley School

"If my theory of relativity is proven successful, Germany will claim me
as a German and France will declare that I am a citizen of the world. Should my theory prove untrue, France will say that
I am a German and Germany will declare that I am a Jew."  Albert Einstein ( 1879  1955 )
"It followed from the special theory of relativity that mass and energy are both but
different manifestations of the same thing  a somewhat unfamiliar conception for the average mind. Furthermore, the equation
E is equal to m csquared, in which energy is put equal to mass, multiplied by the square of the velocity of light, showed
that very small amounts of mass may be converted into a very large amount of energy and vice versa. The mass and energy were
in fact equivalent, according to the formula mentioned above. This was demonstrated by Cockcroft and Walton in 1932, experimentally."
 Prof. Albert Einstein ( excerpts from 1947 film, "Atomic Physics" )
source: American Institute of Physics ( http://www.aip.org/history/einstein/ )
Special Relativity was first published in 1905 by Albert Einstein at age 26 working quietly in the Swiss Patent
Office, Bern, Switzerland, under the title "On The Electrodynamics Of Moving Bodies", translated from "Zur Elektrodynamik bewegter Körper", Annalen der Physik, volume 17: 891, 1905, a downloadable
copy of which is available here in pdf.
Also read "On The Relativity Principle And The Conclusions Drawn From It", by A. Einstein, translated from Jahrbuch der Radioaktivität und Elektronik volume 4 (1907): 411462
And, "Does the Inertia of a Body Depend upon its EnergyContent?", by A. Einstein, Annalen der Physik volume 18: 639, 1905
§ Define:
§ Some Derivations:
1).
.
2).
§ The Problem:
However the entire classical Newtonian physics derived above is predicated upon the concept of mass as an invariant
constant. But we now know differently, namely that mass, m, is a variable quantity owing to the Addition of Relativistic Velocities, where
is the relationship between rest mass undergoing velocity and its equivalent dilated mass.
§ The Solution:
But, whoa! Look,
§ More Simple Algebraic Derivation:
note: see another quick and dirty matheamtical derivation
§ Einstein's Interpretation:
The interpretation that Einstein therefore applied is as follows:
Nevertheless it still should always be remembered that
On the other hand, applying a relativistic kinetic energy concept, we can arrive at the following:
§ The Law of Inertia of Energy:
Or,
is the equation for matter in the form of inertial ( dilated ) mass which can be derived from a given amount
of energy E whose capability for performing work is given by
.
§ Derivation of classical Newtonian kinetic energy:
§ Derivation of relativistic energy:
However for , relativistic mass dilation as a function of velocity,
where is rest mass ( proper mass ) within a given inertial frame of reference, we still have
However using a dummy variable trick for integrating,
And therefore,
§ 2nd Derivation of relativistic energy:
§ Derivation of classical kinetic energy:
§ Here we now have these important energy definitions:
Therefore,
§ Derive the law of conservation of total energy, relativistic and non  relativistic:
Finally, using the binomial series to derive the law of conservation of total energy, relativistic and non  relativistic:
§ Deriving mass dilation using Richard Fehnman's suggested equations from his "Lectures on Physics  Vol. I
" ( although this derivation is somewhat recursive ):
"Imagination is more important than knowledge"  Albert Einstein ( 1879
 1955 )
